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Flag transitive Steiner Systems after Michael Huber Francis Buekenhout Université libre de Bruxelles Académie Royale de Belgique Classe des Sciences Lecture for Finite Geometry Irsee 13-09-06

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Michael Huber (1972- Universität Tübingen

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Steiner System S=S(t,k,v) Finite set of Points equipped with subsets called Blocks There are v points. Each block has k points Every set of t points in a unique block

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Derived Steiner System S=S(t,k,v) Fix point p Then S p, consting of all points other than p and all blocks on p with p deleted, is a S(t-1, k-1, v-1)

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Huber’s Lemma Aut S is 2-transitive on points S flag transitive, p a point In S p, (Aut S) p is transitive on the blocks By Block’s Lemma it is transitive on the points So, Aut S is 2-transitive on the points FLAG TRANSITIVE implies 2- TRANSITIVE

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Non-trivial Steiner System We assume t≥3 but t= 1 or 2 is not bad Forget t=k Forget k=v So 2

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Flag transitive S Flag of S: pair (p, B) where p is a point and B is a block containing p Aut S: group of automorphisms of S S flag transitive: Aut S acts transitively on the flags of S

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Block’s Lemma 1965 R.E. Block, Transitive groups of collineations on certain designs, Pacific J. Math. 15, 1965, 13-18 If G is a group of automorphisms of a rank 2 incidence structure and the number of points is smaller or equal to the number of blocks. Then the number of point orbits of G is not larger than the number of line orbits of G Consequence: if G is block transitive then G is point transitive

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Huber’s Deep Theorem IF S=S(t,k,v) flag transitive, non-trivial, 3≤ t THEN S is known Observe: this remains true if you remove « non- trivial ». It remains also true for t=2 (by BDDKLS) except for an open case.

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About the proof 1 First came Cameron-Praeger For t≥ 7, the result is due to Cameron- Praeger (In Deinze 1992) P.J. Cameron and C. E. Praeger, Block - transitive t-designs, II, large t. In Finite Geometry and Combinatorics (Deinze 1992), Editors: F. De Clerck, e.a. London Math. Soc. Lecture Note Series 191, Cambridge U.P., 1993, 103-119.

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About the proof 2 Apply Huber’s Lemma This is leaving four cases namely t=3, t=4, t=5, t=6. By Huber’s Lemma, Aut S is 2-transitive on the points of S

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About the Proof 3 Apply classification of 2-transitive groups The 2-transitive permutation groups are known. This is due to combined work of Curtis, Kantor, Seitz ( 1976) Gorenstein (1982) Hering (1974, 1985) Huppert (1957) Kantor (1985) Maillet (1895)

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About the Proof 4 Apply classification of 2-transitive groups The 2-transitive permutation groups are known. They appear in a list of 21 classes or types. Task of Huber now: deal with 84 cases. In each case he knows: t, the number v of points, the automorphism group G and its action on S, in particular the structure of a point stabilizer. The size k of a block and the nature of a block remain unknown. Huber’s miracle: the difficulty can be overcome.

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A great ancestor: Heinz Lüneburg 1965 The work of Lüneburg has been a truly inspiring source for Huber. H. Lüneburg, Fahnenhomogenen Quadrupelsysteme, Math. Zeit. 89, 1965, 82-90 M. Huber, Classification of flag-transitive Steiner Quadruple Systems, JCT(A), 94, 2001, 180-190

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A great ancestor: Jacques Tits (Roma 1963) J. Tits, Sur les systèmes de Steiner associés aux trois « grands » groupes de Mathieu, Rendiconti di Mat. 23, 1964,166-184 This remained unknown to Huber until recently Tits uses a concept of t+1 independent points namely t+1 points not on a block (that I would call an apartment)

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A great ancestor: Jacques Tits (Roma 1963) 2 Theorem 1 (Tits) IF S is transitive on ordered apartments and any two blocks intersecting in t-2 points at least do intersect in t-1 points, THEN S is known. Huber generalizes this result. Theorem 2 of Tits is also generalized by Huber.

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Huber’s list 1 (1) Affine space. Here, t=3, v=2 d, k=4, points and planes of AG(d, 2) and one of (1.1) d≥3 and G= AGL(d,2) (1.2) d=3 and G=AGL(1,8) or AG*L(1,8) (1. 3) d=4 and G 0 =A 7 (1.4) d=5 and G=AG*L(1,32) G* means gamma

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Huber’s list 2 (2) Projective line. Here, t=3, v=q e +1, k=q+1, q is a prime power, q≥3, e is an integer, e≥2. Points are those from the projective line over GF(q e ). Blocks are the sublines over GF(q). Also, PSL(2,q) ≤ G≤ PG*L(2,q) and it is allowed that G is PSL(2,q) for e odd.

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Huber’s list 3 (3) Extended Netto System. Here t=3, v=q+1, k=4, the points are those of a projective line over GF(q) where q is a prime power with q=7(mod 12), the blocks are the transforms under PSL(2,q) of {0, 1, e, infinity}where e is a primitive sixth root of unity in GF(q). Also PSL(2,q) ≤ G ≤ PS*L(2,q)

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Huber’s list 4 (4) Witt-Mathieu system. Here, one of the following holds: (4.1) t=3, v=22, k=6 G=M 22 or M 22.2 (4.2) t=4, v=11, k=5 G=M 11 (4.3) t=4, v=23, k=7 G=M 23 (4.4) t=5, v=12, k=6 G=M 12 (4.5) t=5, v=24, k=8 G=M 24 or PSL(2, 23)

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